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📰 Solution: First, calculate the total number of ways to choose 5 opportunities from 8 without restrictions: $\binom{8}{5} = 56$. Next, subtract the number of invalid portfolios that include both mutually exclusive opportunities. If both are selected, the remaining 3 are chosen from the other 6: $\binom{6}{3} = 20$. Thus, the valid portfolios are $56 - 20 = 36$. The final answer is $\boxed{36}$. 📰 Question: A science educator designs a 6-module interactive curriculum using 3 physics simulations, 2 chemistry experiments, and 1 biology lab. How many distinct arrangements are possible if the biology lab must not be placed after the chemistry experiments? 📰 Solution: First, arrange the 6 modules without restrictions: $\frac{6!}{3!2!1!} = 60$. For the constraint, note the biology lab (B) must not follow both chemistry experiments (C). Total valid arrangements: Calculate total permutations where B is not after both C's. This is equivalent to ensuring B is not in a position after both C's. Using combinatorial cases: B is first, or B is second with at least one C before it, or B is third with at least two C's before it. Alternatively, recognize that the condition excludes only $ \frac{1}{3} $ of all permutations where B is after both C's (since the C's can be ordered in 2 ways). Thus, valid arrangements: $60 - \frac{1}{3} \times 60 = 40$. The final answer is $\boxed{40}$. 📰 Emma Is Designing A Rectangular Garden Where The Length Is 3 Times The Width If The Perimeter Of The Garden Is 64 Meters What Are The Dimensions Of The Garden 2692134 📰 What Is Marginal Cost 1607930 📰 Almu Stock Price 3509341 📰 22950 60881 📰 Bank Of America Newbury Park Ca 8603238 📰 Why Everyone Is Talking About Sasha Prasads Hidden Strategythe Shocking Revelation 6609347 📰 1899 Tv Series 613694 📰 Game Studios Careers 3218048 📰 Top Rated Tv Shows You Cant Missgarantised Hits Youll Love Today 1341781 📰 San Francisco Airport Code 9732310 📰 Indiana Food 6909278 📰 Endoplasmic Reticulum Diagram 5475332 📰 John Pork Is Speakingyou Wont Believe What He Just Said 456231 📰 2014 Gmc Acadia 7683101 📰 Government Offices 4192635